Exercise 2 (5 points)

An online platform offers two types of video games: a game of type $A$ and a game of type $B$.

Part A

The durations of games of type $A$ and type $B$, expressed in minutes, can be modeled respectively by two random variables $X_A$ and $X_B$. The random variable $X_A$ follows the uniform distribution on the interval $[9; 25]$. The random variable $X_B$ follows the normal distribution with mean $\mu$ and standard deviation 3.

1. a. Calculate the average duration of a game of type $A$. b. Specify using the graph the average duration of a game of type $B$.
2. We choose at random, with equal probability, a game type. What is the probability that the duration of a game is less than 20 minutes? Give the result rounded to the nearest hundredth.

Part B

It is admitted that, as soon as the player completes a game, the platform proposes a new game according to the following model:

• if the player completes a game of type $A$, the platform proposes to play again a game of type $A$ with probability 0.8;
• if the player completes a game of type $B$, the platform proposes to play again a game of type $B$ with probability 0.7.

For a natural number $n$ greater than or equal to 1, we denote $A_n$ and $B_n$ the events: $A_n$: ``the $n$-th game is a game of type $A$.'' $B_n$: ``the $n$-th game is a game of type $B$.'' For any natural number $n$ greater than or equal to 1, we denote $a_n$ the probability of event $A_n$.

1. a. Copy and complete the probability tree. b. Show that for any natural number $n \geqslant 1$, we have: $a_{n+1} = 0.5\,a_n + 0.3$.

In the rest of the exercise, we denote $a$ the probability that the player plays game $A$ during his first game, where $a$ is a real number belonging to the interval $[0; 1]$. The sequence $(a_n)$ is therefore defined by: $a_1 = a$, and for any natural number $n \geqslant 1$, $a_{n+1} = 0.5\,a_n + 0.3$.

1. Study of a particular case. In this question, we assume that $a = 0.5$. a. Show by induction that for any natural number $n \geqslant 1$, we have: $0 \leqslant a_n \leqslant 0.6$. b. Show that the sequence $(a_n)$ is increasing. c. Show that the sequence $(a_n)$ is convergent and specify its limit.
2. Study of the general case. In this question, the real number $a$ belongs to the interval $[0; 1]$. We consider the sequence $(u_n)$ defined for any natural number $n \geqslant 1$ by $u_n = a_n - 0.6$. a. Show that the sequence $(u_n)$ is a geometric sequence. b. Deduce that for any natural number $n \geqslant 1$, we have: $a_n = (a - 0.6) \times 0.5^{n-1} + 0.6$. c. Determine the limit of the sequence $(a_n)$. Does this limit depend on the value of $a$? d. The platform broadcasts an advertisement inserted at the beginning of games of type $A$ and another inserted at the beginning of games of type $B$. Which advertisement should be the most viewed by a player intensively playing video games?

Exercise 1 — 7 points

Topics: Probability

In Hugo's shop, customers can rent two types of bicycles: road bikes or mountain bikes. Each type of bicycle can be rented in an electric version or not.
A customer is chosen at random from the shop, and we assume that:

• If the customer rents a road bike, the probability that it is an electric bike is 0.4;
• If the customer rents a mountain bike, the probability that it is an electric bike is 0.7;
• The probability that the customer rents an electric bike is 0.58.

We denote by $\alpha$ the probability that the customer rents a road bike, with $0 \leqslant \alpha \leqslant 1$. We consider the following events:

• R: ``the customer rents a road bike'';
• $E$ : ``the customer rents an electric bike'';
• $\bar { R }$ and $\bar { E }$, complementary events of $R$ and $E$.

We model this random situation using the tree shown below. If $F$ denotes any event, we denote by $p ( F )$ the probability of $F$.

1. Copy this tree onto your answer sheet and complete it.
2. a. Show that $p ( E ) = 0.7 - 0.3 \alpha$. b. Deduce that: $\alpha = 0.4$.
3. We know that the customer rented an electric bike. Determine the probability that they rented a mountain bike. Give the result rounded to the nearest hundredth.
4. What is the probability that the customer rents an electric mountain bike?
5. The daily rental price of a non-electric road bike is 25 euros, that of a non-electric mountain bike is 35 euros. For each type of bike, choosing the electric version increases the daily rental price by 15 euros. We denote by $X$ the random variable modeling the daily rental price of a bike. a. Give the probability distribution of $X$. Present the results in the form of a table. b. Calculate the expected value of $X$ and interpret this result.
6. When 30 of Hugo's customers are chosen at random, we treat this choice as sampling with replacement. We denote by $Y$ the random variable associating to a sample of 30 randomly chosen customers the number of customers who rent an electric bike. We recall that the probability of event $E$ is: $p ( E ) = 0.58$. a. Justify that $Y$ follows a binomial distribution and specify its parameters. b. Determine the probability that a sample contains exactly 20 customers who rent an electric bike. Give the result rounded to the nearest thousandth. c. Determine the probability that a sample contains at least 15 customers who rent an electric bike. Give the result rounded to the nearest thousandth.

Exercise 4
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
A museum offers visits with or without an audioguide. Tickets can be purchased online or directly at the counter.

1. When a person buys their ticket online, a validation code is sent to them by SMS so they can confirm their purchase. This code is generated randomly and consists of 4 digits that are pairwise distinct, with the first digit being different from 0.
   Statement 1: The number of different codes that can be generated is 5040.
   
2. A study made it possible to consider that:
   • the probability that a person chooses the audioguide given that they bought their ticket online is equal to 0{,}8;
   • the probability that a person buys their ticket online is equal to 0{,}7;
   • the probability that a person opts for a visit without an audioguide is equal to 0{,}32.
   
   Statement 2: The probability that a visitor does not take the audioguide given that they bought their ticket at the counter is greater than two thirds.
   
3. We randomly choose 12 visitors to this museum.
   We assume that the choice of the ``audioguide'' option is independent from one visitor to another.
   Statement 3: The probability that exactly half of these visitors opt for the audioguide is equal to $924 \times 0{,}2176^6$.
   
4. When a person has an audioguide, they can choose from three routes:
   • a first one lasting fifty minutes,
   • a second one lasting one hour and twenty minutes,
   • a third one lasting one hour and forty minutes.
   
   The tour time can be modelled by a random variable $X$ whose probability distribution is given below:
   

   $x_i$	$50\,\min$	$1\,\mathrm{h}\,20\,\min$	$1\,\mathrm{h}\,40\,\min$
   $P(X = x_i)$	0{,}1	0{,}6	0{,}3

   
   Statement 4: The expectation of $X$ is 77 minutes.

(a) $n$ identical chocolates are to be distributed among the $k$ students in Tinku's class. Find the probability that Tinku gets at least one chocolate, assuming that the $n$ chocolates are handed out one by one in $n$ independent steps. At each step, one chocolate is given to a randomly chosen student, with each student having equal chance to receive it.
(b) Solve the same problem assuming instead that all distributions are equally likely. You are given that the number of such distributions is $\binom { n + k - 1 } { k - 1 }$. (Here all chocolates are considered interchangeable but students are considered different.)

There is a regular tetrahedron-shaped box with the numbers $1,1,1,2$ written one on each face. When this box is thrown, if the number on the bottom face is 1, region A in the figure on the right is colored, and if the number is 2, region B is colored. When the box is thrown repeatedly until both regions are colored, find the probability that the process is completed on the 3rd throw. If this probability is $\frac { q } { p }$, find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

There is a regular tetrahedron-shaped box with the numbers $1,1,1,2$ written one on each face. When this box is thrown, if the number on the bottom face is 1, color region A in the figure on the right; if the number is 2, color region B. Continue throwing this box until both regions are colored. Find the probability that the process is completed on the 3rd throw, expressed as $\frac { q } { p }$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]

A bag contains 1 ball with the number 1, 2 balls with the number 2, and 5 balls with the number 3. One ball is randomly drawn from the bag, the number on the ball is confirmed, and then it is returned. This trial is repeated 2 times. Let $\bar { X }$ be the average of the numbers on the drawn balls. What is the value of $\mathrm { P } ( \bar { X } = 2 )$? [4 points]
(1) $\frac { 5 } { 32 }$
(2) $\frac { 11 } { 64 }$
(3) $\frac { 3 } { 16 }$
(4) $\frac { 13 } { 64 }$
(5) $\frac { 7 } { 32 }$

A die is rolled 5 times, and let $a$ be the number of times an odd number appears. A coin is tossed 4 times, and let $b$ be the number of times heads appears. If the probability that $a - b = 3$ is $\frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [3 points]

A bag contains 5 balls labeled with the numbers $3, 3, 4, 4, 4$, one each. Using this bag and one die, a trial is performed to obtain a score according to the following rule:
If the ball drawn from the bag is labeled 3, roll the die 3 times and the sum of the three results is the score. If the ball drawn from the bag is labeled 4, roll the die 4 times and the sum of the four results is the score.
What is the probability that the score obtained from one trial is 10 points? Express this as $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [1 point]

There is a bag containing 5 balls with the numbers $3,3,4,4,4$ written on them, one each. Using this bag and one die, a trial is performed to obtain a score according to the following rule.
A ball is randomly drawn from the bag. If the number on the drawn ball is 3, the die is rolled 3 times and the sum of the three numbers shown is the score. If the number on the drawn ball is 4, the die is rolled 4 times and the sum of the four numbers shown is the score.
What is the probability that the score obtained from one trial is 10 points? [4 points]
(1) $\frac { 13 } { 180 }$
(2) $\frac { 41 } { 540 }$
(3) $\frac { 43 } { 540 }$
(4) $\frac { 1 } { 12 }$
(5) $\frac { 47 } { 540 }$

15. Teams A and B are playing a best-of-seven basketball series (the series ends when one team wins four games). Based on previous results, Team A's home and away arrangement is ``home, home, away, away, home, away, home'' in order. The probability that Team A wins at home is 0.6, and the probability that Team A wins away is 0.5. Each game is independent. The probability that Team A wins 4-1 is $\_\_\_\_$.

15. Teams A and B are in a basketball championship series using a best-of-seven format (the series ends when one team wins four games). Based on previous results, Team A's home/away schedule is ``home, home, away, away, home, away, home'' in order. Team A's probability of winning at home is 0.6, and away is 0.5. Each game is independent. The probability that Team A wins 4-1 is $\_\_\_\_$ .

18. (12 points) In an 11-point table tennis match, each point won scores 1 point. When the score reaches 10:10, players alternate serves, and the first player to score 2 more points wins the match. Two students, A and B, play a singles match. Assume that when A serves, A scores with probability 0.5; when B serves, A scores with probability 0.4. The results of each point are independent. After a certain match reaches 10:10 with A serving first, the two players play $X$ more points before the match ends.
(1) Find $P ( X = 2 )$;
(2) Find the probability of the event ``$X = 4$ and A wins''.

In the particular case $n = 2$, $m _ { 11 } , m _ { 12 } , m _ { 21 }$ and $m _ { 22 }$ are four real random variables, mutually independent, all following the distribution $\mathcal { R }$ and $M _ { 2 } = \left( \begin{array} { l l } m _ { 11 } & m _ { 12 } \\ m _ { 21 } & m _ { 22 } \end{array} \right)$.
Calculate the probability of the event $M _ { 2 } \in \mathcal { N } _ { 2 }$.

In the particular case $n = 2$, $m _ { 11 } , m _ { 12 } , m _ { 21 }$ and $m _ { 22 }$ are four real random variables, mutually independent, all following the distribution $\mathcal { R }$ and $M _ { 2 } = \left( \begin{array} { l l } m _ { 11 } & m _ { 12 } \\ m _ { 21 } & m _ { 22 } \end{array} \right)$.
Calculate the probability of the event $M _ { 2 } \in \mathcal { G } \ell _ { 2 } ( \mathbb { R } )$.

We consider $2n$ real random variables $c _ { 1 } , c _ { 2 } , \ldots , c _ { n }$ and $c _ { 1 } ^ { \prime } , c _ { 2 } ^ { \prime } , \ldots , c _ { n } ^ { \prime }$ that are mutually independent, all following the distribution $\mathcal { R }$.
Let $\left( \varepsilon _ { 1 } , \ldots , \varepsilon _ { n } \right) \in \{ - 1,1 \} ^ { n }$. Calculate $\mathbb { P } \left( \left( c _ { 1 } = \varepsilon _ { 1 } \right) \cap \cdots \cap \left( c _ { n } = \varepsilon _ { n } \right) \right)$.

We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. For every $n \in \mathbf{N}^*$, we denote $S_n = \sum_{k=1}^n X_k$. We fix the integer $n \geqslant 1$. A path is any $2n$-tuple $\gamma = (\varepsilon_1, \cdots, \varepsilon_{2n})$ whose components $\varepsilon_k$ equal $-1$ or $1$. An equality index of a path is any integer $k \in \llbracket 1, 2n \rrbracket$ such that $\sum_{i=1}^k \varepsilon_i = 0$. For every integer $i$ between $1$ and $n$, the event $A_i$ is defined by: $$A_i = \left\{\omega,\; 2i \text{ is an equality index of } (X_1(\omega), \cdots, X_{2n}(\omega))\right\}.$$ Calculate the probability $\mathbf{P}(A_i)$, for every integer $i$ between $1$ and $n$.

We consider a sequence of random variables $(X_n : \Omega \longrightarrow \{-1,1\})_{n \in \mathbf{N}}$ defined on the same probability space $(\Omega, \mathscr{A}, P)$, taking values in $\{-1,1\}$, mutually independent and centered. For every $n \in \mathbf{N}^*$, we denote $S_n = \sum_{k=1}^n X_k$. Let $\ell \in \mathbf{Z}$ be an integer and $n \geqslant 1$ be another integer. By distinguishing the case where the integer $\ell - n$ is even or odd, calculate $\mathbf{P}(S_n = \ell)$.

Football teams $T _ { 1 }$ and $T _ { 2 }$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T _ { 1 }$ winning, drawing and losing a game against $T _ { 2 }$ are $\frac { 1 } { 2 } , \frac { 1 } { 6 }$ and $\frac { 1 } { 3 }$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T _ { 1 }$ and $T _ { 2 }$, respectively, after two games.
$P ( X > Y )$ is
(A) $\frac { 1 } { 4 }$
(B) $\frac { 5 } { 12 }$
(C) $\frac { 1 } { 2 }$
(D) $\frac { 7 } { 12 }$

Football teams $T _ { 1 }$ and $T _ { 2 }$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T _ { 1 }$ winning, drawing and losing a game against $T _ { 2 }$ are $\frac { 1 } { 2 } , \frac { 1 } { 6 }$ and $\frac { 1 } { 3 }$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 point for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T _ { 1 }$ and $T _ { 2 }$, respectively, after two games.
$P ( X = Y )$ is
(A) $\frac { 11 } { 36 }$
(B) $\frac { 1 } { 3 }$
(C) $\frac { 13 } { 36 }$
(D) $\frac { 1 } { 2 }$

A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is $p$. Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colours is $q$. If $p : q = m : n$, where $m$ and $n$ are co-prime, then $m + n$ is equal to

Let $N$ denote the sum of the numbers obtained when two dice are rolled. If the probability that $2^N < N!$ is $\frac{m}{n}$ where $m$ and $n$ are coprime, then $4m - 3n$ is equal to
(1) 6
(2) 12
(3) 10
(4) 8

One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4 . Another die has one face marked 1 , two faces marked 2 , two faces marked 3 and one face marked 4 . The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 4 } { 9 }$
(4) $\frac { 3 } { 5 }$

There are two boxes, A and B.
In box A, there are three cards on which the number 0 is written, two cards on which the number 2 is written, and one card on which the number 3 is written.
In box B, there are two cards on which the number 1 is written, and three cards on which the number 2 is written.
Take two cards together from box A and one card from box B. Denote the product of the numbers on the three cards by $X$.
The total number of values which $X$ can take is A. The maximum value which $X$ can take is $\mathbf{BC}$. The minimum value which $X$ can take is D.
The probability that $X = \mathrm{BC}$ is $\frac{\mathbf{E}}{\mathrm{F}}$, and the probability that $X = \square\mathrm{D}$ is $\frac{\mathbf{H}}{\mathbf{I}}$.

Let us throw one dice three times, and let the number that comes up on the first throw be $a$, on the second throw be $b$, and on the third throw be $c$. Using these $a , b$ and $c$, we consider the quadratic function $f ( x ) = a x ^ { 2 } + b x + c$.
(1) The probability that $b = 4$ and that the quadratic equation $f ( x ) = 0$ has two different real solutions is $\frac { \mathbf { N } } { \mathbf { O P Q } }$.
(2) Let us find the probability that $f ( 10 ) > 453$.
The number of the cases of $( a , b , c )$ such that $f ( 10 ) > 453$ is as follows: when $a = 4$ and $b = 5$, it is $\mathbf { R }$; when $a = 4$ and $b = 6$, it is $\mathbf{S}$; when $a = 5$, it is $\mathbf { T U }$; when $a = 6$, it is $\mathbf{VW}$. Hence, the probability that $f ( 10 ) > 453$ is $\frac { \mathbf { X } } { \mathbf { Y } }$.